| L(s) = 1 | + 1.93·2-s + 1.73·4-s + 2.44·7-s − 0.517·8-s − 11-s + 4.24·13-s + 4.73·14-s − 4.46·16-s + 5.27·17-s − 2·19-s − 1.93·22-s + 2.07·23-s + 8.19·26-s + 4.24·28-s + 9.46·29-s − 7.46·31-s − 7.58·32-s + 10.1·34-s + 4.89·37-s − 3.86·38-s + 9.46·41-s − 6.03·43-s − 1.73·44-s + 3.99·46-s + 2.82·47-s − 1.00·49-s + 7.34·52-s + ⋯ |
| L(s) = 1 | + 1.36·2-s + 0.866·4-s + 0.925·7-s − 0.183·8-s − 0.301·11-s + 1.17·13-s + 1.26·14-s − 1.11·16-s + 1.28·17-s − 0.458·19-s − 0.411·22-s + 0.431·23-s + 1.60·26-s + 0.801·28-s + 1.75·29-s − 1.34·31-s − 1.34·32-s + 1.74·34-s + 0.805·37-s − 0.626·38-s + 1.47·41-s − 0.920·43-s − 0.261·44-s + 0.589·46-s + 0.412·47-s − 0.142·49-s + 1.01·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.245623438\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.245623438\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 1.93T + 2T^{2} \) |
| 7 | \( 1 - 2.44T + 7T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 - 5.27T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 2.07T + 23T^{2} \) |
| 29 | \( 1 - 9.46T + 29T^{2} \) |
| 31 | \( 1 + 7.46T + 31T^{2} \) |
| 37 | \( 1 - 4.89T + 37T^{2} \) |
| 41 | \( 1 - 9.46T + 41T^{2} \) |
| 43 | \( 1 + 6.03T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 0.757T + 53T^{2} \) |
| 59 | \( 1 + 0.928T + 59T^{2} \) |
| 61 | \( 1 - 8.92T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 9.14T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 + 7.07T + 83T^{2} \) |
| 89 | \( 1 - 8.53T + 89T^{2} \) |
| 97 | \( 1 - 3.58T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.714873215404794875474442346195, −8.177079794864769404774393803224, −7.24410577527560479997189824182, −6.29954450941397601481191338375, −5.63276669298216188616001364069, −4.97321041744584758264870506325, −4.17696466558021251588437295365, −3.41618893920323341835173071046, −2.46914966941170226930684524310, −1.17448457215521381063470044501,
1.17448457215521381063470044501, 2.46914966941170226930684524310, 3.41618893920323341835173071046, 4.17696466558021251588437295365, 4.97321041744584758264870506325, 5.63276669298216188616001364069, 6.29954450941397601481191338375, 7.24410577527560479997189824182, 8.177079794864769404774393803224, 8.714873215404794875474442346195
